Squeezing light through nanometre-wide gaps in metals can lead to extreme field enhancements, nonlocal electromagnetic effects and light-induced electron tunnelling. This intriguing regime, however, has not been readily accessible to experimentalists because of the lack of reliable technology to fabricate uniform nanogaps with atomic-scale resolution and high throughput. Here we introduce a new patterning technology based on atomic layer deposition and simple adhesive-tape-based planarization. Using this method, we create vertically oriented gaps in opaque metal films along the entire contour of a millimetre-sized pattern, with gap widths as narrow as 9.9 Å, and pack 150,000 such devices on a 4-inch wafer. Electromagnetic waves pass exclusively through the nanogaps, enabling backgroundfree transmission measurements. We observe resonant transmission of near-infrared waves through 1.1-nm-wide gaps (l/1,295) and measure an effective refractive index of 17.8. We also observe resonant transmission of millimetre waves through 1.1-nm-wide gaps (l/4,000,000) and infer an unprecedented field enhancement factor of 25,000.

Plasmonic nanoantennas are versatile tools for coherently controlling and directing light on the nanoscale. For these antennas, current fabrication techniques such as electron beam lithography (EBL) or focused ion beam (FIB) milling with Ga(+)-ions routinely achieve feature sizes in the 10 nm range. However, they suffer increasingly from inherent limitations when a precision of single nanometers down to atomic length scales is required, where exciting quantum mechanical effects are expected to affect the nanoantenna optics. Here, we demonstrate that a combined approach of Ga(+)-FIB and milling-based He(+)-ion lithography (HIL) for the fabrication of nanoantennas offers to readily overcome some of these limitations. Gold bowtie antennas with 6 nm gap size were fabricated with single-nanometer accuracy and high reproducibility. Using third harmonic (TH) spectroscopy, we find a substantial enhancement of the nonlinear emission intensity of single HIL-antennas compared to those produced by state-of-the-art gallium-based milling. Moreover, HIL-antennas show a vastly improved polarization contrast. This superior nonlinear performance of HIL-derived plasmonic structures is an excellent testimonial to the application of He(+)-ion beam milling for ultrahigh precision nanofabrication, which in turn can be viewed as a stepping stone to mastering quantum optical investigations in the near-field.

In this paper, we derive a governing equation for spectral asymmetry in electromagnetically induced transparency (EIT). From the key parameters of asymmetry factor - namely dark mode quality factor Q(d), and frequency separation between bright and dark mode Δω(bd) = (ω(b) - ω(d)) -, a logical pathway for the maximization of EIT asymmetry is identified. By taking the plasmonic metal-insulator-metal (MIM) waveguide as a platform, a plasmon-induced transparency (PIT) structure of tunable frequency separation Δω(bd) and dark mode quality factor Q(d) is suggested and analyzed. Compared to previous works on MIM-based plasmon modulators, an order of increase in the performance Fig. (12dB contrast at ~60% throughput) was achieved from the highly asymmetric, narrowband PIT spectra.

Non-Euclidean geometry, discovered by negating Euclid's parallel postulate, has been of considerable interest in mathematics and related fields for the description of geographical coordinates, Internet infrastructures, and the general theory of relativity. Notably, an infinite number of regular tessellations in hyperbolic geometry-hyperbolic lattices-can extend Euclidean Bravais lattices and the consequent band theory to non-Euclidean geometry. Here we demonstrate topological phenomena in hyperbolic geometry, exploring how the quantized curvature and edge dominance of the geometry affect topological phases. We report a recipe for the construction of a Euclidean photonic platform that inherits the topological band properties of a hyperbolic lattice under a uniform, pseudospin-dependent magnetic field, realizing a non-Euclidean analogue of the quantum spin Hall effect. For hyperbolic lattices with different quantized curvatures, we examine the topological protection of helical edge states and generalize Hofstadter's butterfly, showing the unique spectral sensitivity of topological immunity in highly curved hyperbolic planes. Our approach is applicable to general non-Euclidean geometry and enables the exploitation of infinite lattice degrees of freedom for band theory.Band theory in condensed-matter physics and photonics, which provides a general picture of the classification of a phase of matter, has been connected to the concept of topology [1][2][3][4] . The discovery of a topologically nontrivial state in a crystalline insulator, having a knotted property of a wavefunction in reciprocal space, has revealed a new phase of electronic 3 , photonic 4 , and acoustic 5 matter: the topological insulator. This topological phase offers the immunity of electronic conductance 1,3 or light transport 2,6-8 against disorder, occurring only on the boundary of the matter through topologically protected edge states.A fundamental approach for generalizing band theory and its topological extension is to rethink the traditional assumptions regarding energy-momentum dispersion relations, such as static, Hermitian, and periodic conditions. In photonics, dynamical lattices can derive an effective magnetic field for photons, realizing one-way edge states in space-time Floquet bands 9 .Non-Hermitian photonics introduces novel band degeneracies 10 and topological phenomena 11 in complex-valued band structures. Various studies have extended band theory to disordered systems, achieving perfect bandgaps in disordered structures 12,13 and topological invariants in amorphous media [14][15][16] .Meanwhile, most of the cornerstones in band theory have employed Euclidean geometry, because Bloch's theorem is well defined for a crystal, which corresponds to the uniform tessellation of Euclidean geometry. However, significant lattice degrees of freedom are overlooked in wave phenomena confined to Bloch-type Euclidean geometry because of the restricted inter-elemental relationships from a finite number of uniform Euclidean tessellations.

We use coupled mode theory (CMT) to analyze a metal-insulator-metal (MIM) plasmonic stub structure, to reveal the existence of asymmetry in its transmittance spectra. Including the effect of the near field contribution for the stub structure, the observed asymmetry is interpreted as Fano-type interference between the quasi-continuum T-junction-resonator local-modes and discrete stub eigenmodes. Based on the asymmetry factor derived from the CMT analysis, methods to control transmittance asymmetry are also demonstrated.